Question: $\overline{AB}$ = $\sqrt{89}$ $\overline{AC} = {?}$ $A$ $C$ $B$ $\sqrt{89}$ $?$ $ \sin( \angle ABC ) = \frac{8\sqrt{89} }{89}, \cos( \angle ABC ) = \frac{5\sqrt{89} }{89}, \tan( \angle ABC ) = \dfrac{8}{5}$
Solution: $\overline{AB}$ is the hypotenuse $\overline{AC}$ is opposite to $\angle ABC$ SOH CAH TOA We know the hypotenuse and need to solve for the opposite side so we can use the sine function (SOH) $ \sin( \angle ABC ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\overline{AC}}{\overline{AB}}= \frac{\overline{AC}}{\sqrt{89}} $ $ \overline{AC}=\sqrt{89} \cdot \sin( \angle ABC ) = \sqrt{89} \cdot \frac{8\sqrt{89} }{89} = 8$